Lecture 1 Review of Statistics

8/2/2019 Lecture 1 Review of Statistics

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ECON 107

Introductory Econometrics

Summer 2011


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Brief Overview of the Course

Economics suggests important relationships, often with policy

implications, but virtually never suggests quantitative

magnitudes of causal effects.

What is the quantitative effect of reducing class size onstudent achievement?

How does another year of education change earnings?What is the price elasticity of cigarettes?What is the effect on output growth of a 1 percentage

point increase in interest rates by the Fed?

What is the effect on housing prices of environmentalimprovements?


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This course is about using data to measure causal effects.

Ideally, we would like an experimentowhat would be an experiment to estimate the effect of

class size on standardized test scores?

But almost always we only have observational(nonexperimental) data.

oreturns to educationocigarette pricesomonetary policy


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In this course you will:

Learn methods for estimating causal effects usingobservational data

Focus on applicationstheory is used only as needed tounderstand the whys of the methods;

Learn to evaluate the regression analysis of othersthismeans you will be able to read/understand empirical

economics papers in other econ courses;

Get some hands-on experience with regression analysis inyour problem sets.


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Review of Statistical Theory

1. The probability framework for statistical inference2. Estimation3. Testing4. Confidence Intervals


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Population distribution of Y

The probabilities of different values ofYthat occur in thepopulation, for ex. Pr[Y= 650] (when Yis discrete)

or: The probabilities of sets of these values, for ex. Pr[640Y 660] (when Yis continuous).


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(b) Moments of a population distribution: mean, variance,

standard deviation, covariance, correlation

mean = expected value (expectation) ofY

=E(Y)

= Y

= long-run average value ofYover repeated

realizations ofY

variance =E(YY)2

= 2Y

= measure of the squared spread of thedistribution

standard deviation = variance = Y


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Moments, ctd.

skewness =

3

3

Y

Y

E Y

= measure of asymmetry of a distribution

skewness = 0: distribution is symmetricskewness > ( 3: heavy tails (leptokurtic)


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Two random variables:joint distributions and covariance

Random variablesXandZhave ajoint distributionThecovariance betweenXandZis

cov(X,Z) =E[(XX)(ZZ)] = XZ

The covariance is a measure of the linear associationbetweenXandZ; its units are units ofX units ofZ

cov(X,Z) > 0 means a positive relation betweenXandZIfXandZare independently distributed, then cov(X,Z) = 0

(but not vice versa!!)

The covariance of a r.v. with itself is its variance:cov(X,X) =E[(XX)(XX)] =E[(XX)

2] = 2X


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The covariance between Test Score and STR is negative:

so is thecorrelation


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Thecorrelation coefficient is defined in terms of the

covariance:

corr(X,Z) = cov( , )

var( ) var( )

XZ

X Z

X Z

X Z

= rXZ

1 corr(X,Z) 1corr(X,Z) = 1 mean perfect positive linear associationcorr(X,Z) =1 means perfect negative linear associationcorr(X,Z) = 0 means no linear association


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Thecorrelation coefficient measures linear association


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(c) Conditional distributions and conditional means

Conditional distributions

The distribution ofY, given value(s) of some otherrandom variable,X

Ex: the distribution of test scores, given that STR < 20Conditional expectations and conditional moments

conditional mean = mean of conditional distribution=E(Y|X=x) (important concept and notation)

conditional variance = variance of conditional distributionExample: E(Test scores|STR < 20) = the mean of test

scores among districts with small class sizes

The difference in means is the difference between the means

of two conditional distributions:


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Conditional mean, ctd.

=E(Test scores|STR < 20)E(Test scores|STR 20)

Other examples of conditional means:

Wages of all female workers (Y= wages,X= gender)Mortality rate of those given an experimental treatment (Y

= live/die;X= treated/not treated)

IfE(X|Z) = const, then cov(X,Z) = 0 (not necessarily viceversa however)

The conditional mean is a term for the familiar idea of thegroup mean


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(d) Distribution of a sample of data drawn randomly

from a population: Y1,, Yn

We will assume simple random sampling

Choose an individual (district, entity) at random from thepopulation

Not using the stratified sampling or cluster sampling.Randomness and data

Prior to sample selection, the value ofYis randombecause the individual selected is random

Once the individual is selected and the value ofYisobserved, then Yis just a numbernot random

The data set is (Y1, Y2,, Yn), where Yi = value ofYfor theith

individual (district, entity) sampled


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Distribution of Y1,, Ynunder simple random sampling

Because individuals #1 and #2 are selected at random, thevalue ofY1 has no information content for Y2. Thus:

oY1 and Y2 are independently distributedoY1 and Y2 come from the same distribution, that is, Y1,

Y2 are identically distributed

oThat is, under simple random sampling, Y1 and Y2 areindependently and identically distributed (i.i.d.).

oMore generally, under simple random sampling, {Yi}, i= 1,, n, are i.i.d.

This framework allows rigorous statistical inferences about

moments of population distributions using a sample of data

from that population


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Estimation

Y is the natural estimator of the mean. But:

(a)What are the properties ofY?(b)Why should we use Y rather than some other estimator?

Y1 (the first observation)maybe unequal weightsnot simple averagemedian(Y1, Yn)12

(min

Y +max

Y )


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(a) The sampling distribution ofY

Y is a random variable, and its properties are determined by

thesampling distribution ofY

The individuals in the sample are drawn at random.Thus the values of (Y1,, Yn) are randomThus functions of (Y1,, Yn), such as Y , are random: had

a different sample been drawn, they would have taken ona different value

The distribution ofY over different possible samples ofsize n is called thesampling distribution ofY .

The mean and variance ofY are the mean and variance ofits sampling distribution, E(Y) and var(Y).


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The concept of the sampling distribution underpins all ofeconometrics.


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The sampling distribution ofY, ctd.

Example: Suppose Ytakes on 0 or 1 (aBernoulli random

variable) with the probability distribution,

Pr[Y= 0] = .22, Pr(Y=1) = .78

Then

E(Y) =p1 + (1p)0 =p = .782

Y

=E[YE(Y)]2

=p(1p) [remember this?]

= .78(1.78) = 0.1716

The sampling distribution ofY depends on n.

Consider n = 2. The sampling distribution ofY is,

Pr(Y = 0) = .222

= .0484

Pr(Y = ) = 2.78 .22= .3432

Pr(Y = 1) = .782

= .6084


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The sampling distribution ofY when Yis Bernoulli (p = .78)


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Things we want to know about the sampling distribution:

What is the mean ofY?oIfE(Y) = true = .78, then Y is an unbiasedestimator

of

What is the variance ofY?oHow does var(Y) depend on n (famous 1/n formula)

Does Y become close to when n is large?oLaw of large numbers: Y is aconsistent estimator of

Y appears bell shaped for nlargeis this generallytrue?

oIn fact, Y is approximately normally distributedfor n large (Central Limit Theorem)


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The mean and variance of the sampling distribution ofY

General casethat is, for Yi i.i.d. from any distribution, not

just Bernoulli:

mean: E(Y) =E(1

1 n

i

i

Yn

) =1

1( )

n

i

i

E Yn

=1

1 n

Y

in

= Y

Variance: var(Y) =E[Y E(Y)]2

=E[Y Y]2

=E

2

1

1 n

i Y

i

Yn

=E

2

1

1( )

n

i Y

i

Yn


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so var(Y) =E

2

1

1( )

n

i Y

i

Yn

=1 1

1 1( ) ( )

n n

i Y j Y

i j E Y Y n n

=2

1 1

1( )( )

n n

i Y j Y

i j

E Y Y n

=2

1 1

1cov( , )

n n

i j

i j

Y Yn

= 22

1

1 n

Y

i

n

=2

Y

n


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Mean and variance of sampling distribution ofY, ctd.

E(Y) = Y

var(Y) =2

Y

n

Implications:

1. Y is an unbiasedestimator ofY(that is,E(Y) = Y)2. var(Y) is inversely proportional to n

the spread of the sampling distribution isproportional to 1/ n


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Thus the sampling uncertainty associated with Y isproportional to 1/ n (larger samples, less

uncertainty, but square-root law)


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The sampling distribution ofY whenn is large

For small sample sizes, the distribution ofY is complicated,

but ifn is large, the sampling distribution is simple!

1. As n increases, the distribution ofY becomes more tightlycentered around Y(theLaw of Large Numbers)

2. Moreover, the distribution ofYYbecomes normal (theCentral Limit Theorem)


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TheLaw of Large Numbers:

An estimator isconsistent if the probability that its falls

within an interval of the true population value tends to one

as the sample size increases.

If (Y1,,Yn) are i.i.d. and2

Y


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The Central Limit Theorem (CLT):

If (Y1,,Yn) are i.i.d. and 0 )

H0:E(Y) = Y,0 vs.H1:E(Y) < Y,0 (1-sided,


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Some terminology for testing statistical hypotheses:

p-value= probability of drawing a statistic (e.g. Y) at least as

adverse to the null as the value actually computed with your

data, assuming that the null hypothesis is true.

= probability of rejecting the null hypothesis when it is true.

Thesignificance levelof a test is a pre-specified probability

of incorrectly rejecting the null, when the null is true.

Calculating the p-value based on Y :

p-value =0 ,0 ,0

Pr [| | | |]act H Y Y

Y Y

h actY i th l f Y t ll b d ( d )


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where actY is the value ofY actually observed (nonrandom)

Calculating the p value ctd


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Calculating the p-value, ctd.

To compute thep-value, you need to know the samplingdistribution ofY , which is complicated ifn is small.

Ifn is large, you can use the normal approximation (CLT):p-value =

0 ,0 ,0Pr [| | | |]act

H Y Y Y Y ,

=0

,0 ,0Pr [| | | |]/ /

act

Y YH

Y Y

Y Yn n

=0

,0 ,0Pr [| | | |]

act

Y Y

H

Y Y

Y Y

probability under left+rightN(0,1) tails

whereY

= std. dev. of the distribution ofY = Y/ n .

C l l ti th l ith k


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Calculating the p-value with Yknown:

For large n,p-value = the probability that aN(0,1) randomvariable falls outside |( actY Y,0)/ Y |

In practice,Y

is unknownit must be estimated

Estimator of the variance of Y:


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Estimator of the variance of Y:

2

Ys = 2

1

1( )

1

n

i

i

Y Y

n

= sample variance ofY

Fact:

If (Y1,,Yn) are i.i.d. andE(Y4)


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Computing the p-value with Y estimated:

p-value =0 ,0 ,0

Pr [| | | |]act H Y Y

Y Y ,

=0

,0 ,0Pr [| | | |]

/ /

act

Y Y

H

Y Y

Y Y

n n

0

,0 ,0Pr [| | | |]

/ /

act

Y Y

H

Y Y

Y Y

s n s n

(large n)

so

p-value =0

Pr [| | | |]actH t t (2

Y estimated)

probability under normal tails outside |tact

|

where t=,0

/

Y

Y

Y

s n

(the usual t-statistic)

What is the link between the p value and the significance


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What is the link between thep-value and the significance

level?

The significance level is prespecified. For example, if theprespecified significance level is =5%,

you reject the null hypothesis if |t| 1.96equivalently, you reject ifp (= 0.05).Often, it is better to communicate thep-value than simply

whether a test rejects or notthep-value contains more

information than the yes/no statement about whether the

test rejects.

At this point you might be wondering


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At this point, you might be wondering, ...

What happened to thet-table and the degrees of freedom?

Digression: the Studentt distributionIfYi, i = 1,, n is i.i.d.N(Y,

2

Y ), then the t-statistic has the

Student t-distribution with n1 degrees of freedom.

The critical values of the Student t-distribution is tabulated in

the back of all statistics books. Remember the recipe?

1. Compute the t-statistic2. Compute the degrees of freedom, which is n13.

Look up the 5% critical value

4. If the t-statistic exceeds (in absolute value) thiscritical value, reject the null hypothesis.

Comments on this recipe and the Student t-distribution


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Comments on this recipe and the Studentt-distribution

1. If the sample size is moderate (several dozen) or large(hundreds or more), the difference between the t-

distribution and N(0,1) critical values are negligible. Hereare some 5% critical values for 2-sided tests:

degrees of freedom

(n1)

5% t-distribution

critical value

10 2.23

20 2.09

30 2.04

60 2.00

1.96

Comments on Student t distribution ctd


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Comments on Student t distribution, ctd.

2. So, the Student-tdistribution is only relevant when thesample size is very small; but even in that case, for it to becorrect, you must be sure that the population distribution of

Yis normal. In economic data, the normality assumption is

rarely credible. Here are the distributions of some

economic data.

Do you think earnings are normally distributed?


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Comments on Student t distribution ctd


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Comments on Student t distribution, ctd.

3. [You might not know this.] Consider the t-statistic testingthe hypothesis that two means (groups s, l) are equal:

2 2 ( )s ls l

s l s l

s ss l

n n

Y Y Y Y tSE Y Y

Even if the population distribution ofYin the two groups

is normal, this statistic doesnt have a Student tdistribution!

There is a statistic testing this hypothesis that has a

normal distribution, the pooled variance t-statisticsee

SW (Section 3.6)however the pooled variance t-statistic

is only valid if the variances of the normal distributions

are the same in the two groups Would you expect this to


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are the same in the two groups. Would you expect this to

be true, say, for mens vs. womens wages?

The Student-t distribution summary


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The Student t distribution summary

The assumption that Yis distributedN(Y, 2Y ) is rarelyplausible in practice (Income? Test score?)

For n > 30, the t-distribution andN(0,1) are very close (asn grows large, the tn1 distribution converges toN(0,1))

The t-distribution is an artifact from days when samplesizes were small and computers were people

For historical reasons, statistical software typically usesthe t-distribution to computep-valuesbut this is

irrelevant when the sample size is moderate or large.For these reasons, in this class we will focus on the large-

n approximation given by the CLT

Confidence Intervals


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Confidence Intervals

A 95%confidence intervalfor Y is an interval that contains

the true value ofY in 95% of repeated samples.

Digression: What is random here? The values ofY1,,Yn and

thus any functions of themincluding the confidence

interval. The confidence interval it will differ from one

sample to the next. The population parameter, Y, is not

random, we just dont know it.


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Summary:


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Summary:

From the two assumptions of:

(1) simple random sampling of a population, that is,{Yi, i=1,,n} are i.i.d.

(2) 0


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Empirical problem: Class size and educational output

Policy question: What is the effect on test scores (or someother outcome measure) of reducing class size by one

student per class? By 8 students/class?

The California Test Score Data Set


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Ca a S a a S

All K-6 and K-8 California school districts (n = 420)

Variables:

5th grade test scores (Stanford-9 achievement test,combined math and reading), district average

Student-teacher ratio (STR) = no. of students in thedistrict divided by no. full-time equivalent teachers

Initial look at the data:


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(You should already know how to interpret this table)

This table doesnt tell us anything about the relationship

between test scores and the STR.

Do districts with smaller classes have higher test scores?


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g

Scatterplot of test score v. student-teacher ratio

What does this figure show?

We need to get some numerical evidence on whether districts


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g

with low STRs have higher test scoresbut how?

1. Compare average test scores in districts with low STRsto those with high STRs (estimation)

2. Test the null hypothesis that the mean test scores inthe two types of districts are the same, against the

alternative hypothesis that they differ (hypothesis

testing)

3. Estimate an interval for the difference in the mean testscores, high v. low STR districts (confidence

interval)

Initial data analysis: Compare districts with small (STR


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y p (

20) and large (STR 20) class sizes:

ClassSize

Average score( j

iY )

Standarddeviation (sY)

n

Small 657.4 19.41

n 238

Large 650.0 17.9 182

1. Estimation of = difference between group means2. Test the hypothesis that = 03. Construct aconfidence intervalfor

1. Estimation


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1 2Y Y =small

1small

1n

i

i

Yn

large

1large

1n

i

i

Yn

= 657.4650.0

= 7.4

Is this a large difference in a real-world sense?

Standard deviation across districts = 19.1Difference between 60th and 75th percentiles of test score

distribution is 667.6659.4 = 8.2

This is a big enough difference to be important for schoolreform discussions, for parents, or for a school

committee?

2. Hypothesis testing


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yp g

Difference-in-means test: compute the t-statistic,

2 2 ( )s ls l

s l s l

s ss l

n n

Y Y Y Y t

SE Y Y

(remember this?)

where SE( sY lY) is the standard error of sY lY , the

subscripts s and lrefer to small and large STR districts,

and 2 2

1

1( )

1

sn

s i sis

s Y Yn

(etc.)

Compute the difference-of-means t-statistic:


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Size Y sY n

small 657.4 19.4 238large 650.0 17.9 182

2 2 2 219.4 17.9

238 182

657.4 650.0 7.4

1.83s ls l

s l

s s

n n

Y Y

t

= 4.05

|t| > 1.96, so reject (at the 5% significance level) the null

hypothesis that the two means are the same.

3. Confidence interval


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A 95% confidence interval for the difference between the

means is,

(s

Y l

Y) 1.96SE(s

Y l

Y)

= 7.4 1.961.83 = (3.8, 11.0)

Two equivalent statements:

1. The 95% confidence interval for doesnt include 0;2. The hypothesis that = 0 is rejected at the 5% level.


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What is the effect on students standardized test scores ofa percentage increase in Student-teacher ratio?

To answer the above question, you will learn regression

analysis in the coming classes.

Documents

Lecture 1 Review of Statistics